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A024314 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.

%I A024314 #19 Jan 09 2024 16:06:50
%S A024314 3,9,24,37,81,133,256,413,746,1208,2098,3394,5753,9309,15532,25131,
%T A024314 41499,67147,110122,178181,290890,470670,766068,1239524,2013407,
%U A024314 3257761,5284656,8550753
%N A024314 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023532.
%H A024314 G. C. Greubel, <a href="/A024314/b024314.txt">Table of n, a(n) for n = 1..1000</a>
%H A024314 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,-1,-1,-3,2,1,1,1).
%F A024314 G.f.: x*(3 + 6*x + 6*x^2 - 8*x^3 - 7*x^4 + x^5 - 4*x^6 + 2*x^7)/((1 - x - x^2)*(1 - x^2 - x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
%F A024314 From _G. C. Greubel_, Jan 17 2022: (Start)
%F A024314 a(2*n) = 6*F(2*n+3) + F(2*n+1) - (n+6)*F(n+3) - (n+1)*F(n+1).
%F A024314 a(2*n+1) = 6*F(2*n+2) + F(2*n) - (n+6)*F(n+2) - (n+1)*F(n), where F(n) = A000045(n). (End)
%t A024314 a[n_]:= With[{F=Fibonacci}, 6*F[n+3] +F[n+1] - (1/2)*((1+(-1)^n)*(((n+2)/2 )*LucasL[(n+4)/2] + 5*F[(n+6)/2]) - (1-(-1)^n)*(((n+3)/2)*LucasL[(n+3)/2] +5*F[(n+5)/2] ))];
%t A024314 Table[a[n], {n, 40}] (* _G. C. Greubel_, Jan 17 2022 *)
%o A024314 (Magma)
%o A024314 R<x>:=PowerSeriesRing(Integers(), 40);
%o A024314 Coefficients(R!( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // _G. C. Greubel_, Jan 17 2022
%o A024314 (Sage)
%o A024314 def A024314_list(prec):
%o A024314     P.<x> = PowerSeriesRing(ZZ, prec)
%o A024314     return P( x*(3+6*x+6*x^2-8*x^3-7*x^4+x^5-4*x^6+2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
%o A024314 a=A024314_list(41); a[1:] # _G. C. Greubel_, Jan 17 2022
%Y A024314 Cf. A024312, A024313, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.
%Y A024314 Cf. A000032, A000045, A023532.
%K A024314 nonn,easy
%O A024314 1,1
%A A024314 _Clark Kimberling_